ORIGINAL PAPER

Independent analysis of the radiation risk for leukaemiain children and adults with mortality data (1950–2003)of Japanese A-bomb survivors

Jan Christian Kaiser • Linda Walsh

Received: 28 March 2012 / Accepted: 21 October 2012 / Published online: 4 November 2012

� The Author(s) 2012. This article is published with open access at Springerlink.com

Abstract A recent analysis of leukaemia mortality in

Japanese A-bomb survivors has applied descriptive models,

collected together from previous studies, to derive a joint

excess relative risk estimate (ERR) by multi-model infer-

ence (MMI) (Walsh and Kaiser in Radiat Environ Biophys

50:21–35, 2011). The models use a linear-quadratic dose

response with differing dose effect modifiers. In the present

study, a set of more than 40 models has been submitted to a

rigorous statistical selection procedure which fosters the

parsimonious deployment of model parameters based on

pairwise likelihood ratio tests. Nested models were conse-

quently excluded from risk assessment. The set comprises

models of the excess absolute risk (EAR) and two types of

non-standard ERR models with sigmoidal responses or two

line spline functions with a changing slope at a break point.

Due to clearly higher values of the Akaike Information

Criterion, none of the EAR models has been selected, but

two non-standard ERR models qualified for MMI. The

preferred ERR model applies a purely quadratic dose

response which is slightly damped by an exponential factor

at high doses and modified by a power function for attained

age. Compared to the previous analysis, the present study

reports similar point estimates and confidence intervals (CI)

of the ERR from MMI for doses between 0.5 and 2.5 Sv.

However, at lower doses, the point estimates are markedly

reduced by factors between two and five, although the

reduction was not statistically significant. The 2.5 % per-

centiles of the ERR from the preferred quadratic-exponen-

tial model did not fall below zero risk in exposure scenarios

for children, adolescents and adults at very low doses down

to 10 mSv. Yet, MMI produced risk estimates with a

positive 2.5 % percentile only above doses of some

300 mSv. Compared to CI from a single model of choice,

CI from MMI are broadened in cohort strata with low sta-

tistical power by a combination of risk extrapolations from

several models. Reverting to MMI can relieve the dilemma

of needing to choose between models with largely different

consequences for risk assessment in public health.

Keywords Leukaemia mortality � Radiation risk �A-bomb survivors � Nonlinear dose response �Multi-model inference

Introduction

In a recent analysis of leukaemia mortality in the Japanese

life span study (LSS) cohort of A-bomb survivors, a joint

radiation risk has been derived from a group of several

models by applying the technique of multi-model inference

(MMI) (Walsh and Kaiser 2011). Reduction of bias from

relying on a single model for risk assessment constitutes

the main virtue of MMI. Application of MMI can produce

more reliable point estimates and improves the character-

isation of uncertainties (Burnham and Anderson 2002).

Electronic supplementary material The online version of thisarticle (doi:10.1007/s00411-012-0437-6) contains supplementarymaterial, which is available to authorized users.

J. C. Kaiser (&)

Helmholtz Zentrum Munchen, German Research Centre

for Environmental Health, Institute of Radiation Protection,

85764 Oberschleissheim, Germany

e-mail: christian.kaiser@helmholtz-muenchen.de

L. Walsh

Department Radiation Protection and Health, Federal Office

for Radiation Protection, 85764 Oberschleissheim, Germany

L. Walsh

The Faculty of Medical and Human Sciences,

University of Manchester, Manchester, UK

123

Radiat Environ Biophys (2013) 52:17–27

DOI 10.1007/s00411-012-0437-6

Walsh and Kaiser (2011) have chosen models for a so-

called group of Occam, after a review of the relevant lit-

erature in radio-epidemiology. The group contained those

models which were deemed adequate for joint risk infer-

ence (Hoeting et al. 1999; Kaiser et al. 2012). They were

then ranked according to the Akaike Information Criterion

(AIC) which penalises models with many parameters. A

joint risk estimate is given by the mean of model-specific

estimates with AIC-based weights, and confidence inter-

vals (CI) are calculated by approximate methods.

For the models discussed in Walsh and Kaiser (2011),

parameter parsimony was not always the main concern of

model authors so that highly parametrised models had

received negligible weights in the weighting process. This

intrinsic feature of MMI was criticised by Richardson and

Cole (2012). They argued that models with explanatory

variables which may have an impact on the radiation risk

are not considered adequately. In their reply, Walsh et al.

(2012) cautioned against the use of model parameters

which are not sufficiently supported by the data. Based on

the hypothetical problem posed by Richardson and Cole

(2012), Walsh et al. (2012) illustrated that models, which

contain parameters with weak statistical support, may

cause misleading point estimates of the risk. In other

examples, over-parametrised models may have little

impact on point estimates but can still inflate uncertainty

ranges artificially. This side-effect contorts risk assessment

in radiation protection if an accurate determination of

uncertainties is desired. Such desire is brought forward in

court cases related to compensation claims for detrimental

health effects from occupational radiation exposure (Niu

et al. 2010). For example, decisions in USA courts are

sometimes based on the 99 % CI of the probability of

causation for cancer in a specific organ (Kocher et al.

2008).

Thus, the criterion for the choice of models for MMI in

the study of Walsh and Kaiser (2011) has been changed

here, so that the advice of Walsh et al. (2012) is taken

seriously. Instead of picking models from peer-reviewed

literature without further qualifications, potential candidate

models are now submitted to a rigorous statistical selection

protocol. Such a protocol has been introduced by Kaiser

et al. (2012) and applied to the selection of both descriptive

and mechanistic breast cancer models for joint risk

inference.

All models considered in Walsh and Kaiser (2011)

include a linear-quadratic dose response with different

combinations of explanatory variables such as sex, age at

exposure and attained age to modify the dose response of

the risk. A linear-quadratic response is also preferred in the

LSS studies on leukaemia incidence (Preston et al. 1994)

and mortality (Preston et al. 2004). It is recommended by

committees BEIR VII (NRC 2006), ICRP (Valentin 2007)

and UNSCEAR (2008) after consideration of a sizeable

number of leukaemia risk studies.

Although the linear-quadratic response can be regarded

as the accepted standard in the radio-epidemiology of

leukaemia, a number of non-standard responses have been

tested motivated by earlier investigations. Little et al.

(1999) found that a quadratic-exponential response yielded

optimal fits when applied to LSS leukaemia incidence data.

Preston et al. (1994) applied a model of two linear dose

responses, represented by two line spline functions with a

changing slope at a break point, as an alternative to the

linear-quadratic response. Explicitly, nonlinear dose

responses with sigmoidal forms have also been investi-

gated. They are well-known in toxicology (Hodgson 2010)

and are applied in radiation biology to describe normal

tissue damage, i.e., of the skin (Hall 2006).

It is emphasised here that the choice of candidate

models is on no account exhaustive and that a possible

inclusion of non-standard models into Occam’s group is

mainly justified by goodness-of-fit criteria.

The assignment of weights to risk models is also prac-

tised to transport organ-specific risk estimates from the

LSS cohort to western populations, if no information on the

radiation risk in Caucasian cohorts is available. However,

committees BEIR VII (NRC 2006) and ICRP (Valentin

2007) support different approaches to combine an excess

absolute risk (EAR) model and an excess relative risk

(ERR) model with weights quantified by expert judgement.

In any case, adequate transfer models must provide a good

description of the risk in the population of origin. The

relevance of this statistical criterion for risk transfer con-

cerning leukaemia will be highlighted by the present study.

Past studies of the leukaemia risk at low doses for young

attained ages and ages at exposure were performed for

settlements in the vicinity of nuclear power stations (NPP)

(Laurier et al. 2008; Kaatsch et al. 2008) and to estimate

the proportion of cases induced by computer tomography

(CT) scans (Pearce et al. 2012) or natural background

radiation (Wakeford et al. 2009; Little et al. 2009; Kendall

et al. 2012). Investigations in these fields and, additionally,

ongoing risk assessment for residents near the Japanese

Fukushima Daiichi NPP may benefit from both risk esti-

mates with stronger support of the data and a more com-

prehensive quantification of uncertainties, which are the

aim of the present study.

Materials and methods

Epidemiological data set

The present study is closely related to the study of Walsh

and Kaiser (2011) which used LSS mortality data from

18 Radiat Environ Biophys (2013) 52:17–27

123

1950 to 2000. After it appeared, the LSS data have been

updated with an extended follow-up to 2003 in Report 14

(Ozasa et al. 2012). To provide an analysis with the most

recent data set, all results reported by the present study are

based on LSS Report 14. The updated data set comprises

86,611 subjects, 318 leukaemia deaths (including 22 cases

in 2001–2003) and 3,294,282 person years (including

109,927 person years in 2001–2003). The person-year

weighted means are 22 year for age at exposure, 50 year

for attained age, 58 year for age of cases and 134 mSv for

the weighted dose to bone marrow with a factor of ten for

the relative biological effectiveness (RBE) of neutrons. The

RBE value depends on the radiation field and the detri-

mental health effect under observation. For leukaemia, an

estimation is difficult and produces very large CI (Little

1997; Hunter and Charles 2002). The LSS cohort data in

file lss14.csv are available for download from the website

of the Radiation Effects Research Foundation (RERF) in

Japan (http://www.rerf.or.jp).

The MECAN software package

The analysis has been performed with the MECAN soft-

ware package which is available from the corresponding

author by request (Kaiser 2010). A user manual, regression

control files and an executable to repeat the present anal-

ysis are included. MECAN is executed in a terminal on a

command line under Linux or Windows. To implement risk

models other than those applied here, a minimal knowledge

of the C?? programming language is required. The code

includes the C?? library MINUIT2 (Moneta and James

2010) from CERN which minimises the Poisson likelihood.

Pre-processing of the grouped data, regression, comparison

of observed and expected cases, and simulation of uncer-

tainty intervals can all be performed in one run. The cal-

culation of risk estimates from MMI is automated with

shell script files which contain the set of required

commands.

Results from MECAN for the preferred models of the

present study and of the study by Walsh and Kaiser (2011)

have been cross-checked by independent calculations with

the EPICURE package (Preston et al. 1993). Deviances

from the two packages differed by around 10-3 points.

Relative differences of estimates for model parameters,

Wald-type standard errors and CI from the likelihood

profile fell below 10-2. Relative differences in the entries

of the parameter correlation matrices exceeded one per cent

in some cases.

Baseline model

The model for the baseline mortality rates

h0ðs; c; a; eÞ ¼ expfb0 þ bssþ bcc

þ ba1ln a=55ð Þ þ ba2

ln2 a=55ð Þþ be1

e� 30ð Þ þ be2e� 30ð Þ2g

ð1Þ

applies the same functional form as the models of the

UNSCEAR committee and of Little et al. (2008) (see

Table 8 of Walsh and Kaiser 2011). The parameter b0

represents a constant factor, parameters bs and bc account

for rate differences by sex (males s = -1 and females

s = ?1) and city, i.e. Hiroshima (c = -1) and Nagasaki

(c = ?1). Parameters ba1and ba2

quantify variations of

the rates with attained age. Parameters be1and be2

depend

on age at exposure which for the acute exposure of the

A-bomb survivors serves as a surrogate for dependence on

birth cohort to account for secular trends in baseline rates.

The present baseline model consumes seven adjustable

parameters.

Model selection protocol

The selection protocol of Kaiser et al. (2012) has been

applied here. It starts with step-by-step attempts to optimise

the baseline model in Eq. (1) with exposure-related features

contained in a set of candidate models. Parameters are

added individually or in groups and retained, if the nested

model with the additional parameter(s) survived a likeli-

hood ratio test (LRT) against the model of origin. For

nested models, the difference between their deviances is

v2-distributed (Claeskens and Hjort 2008; Walsh 2007).

The number of degrees of freedom for the difference is

equal to the difference in the number of parameters. A

model with one additional parameter is considered an

improvement over the model without this parameter with a

95 % probability if the deviance is lowered by at least 3.84

points. The probability threshold is set relatively high to

avoid inclusion of spurious features in risk models

(Anderson et al. 2001; Walsh et al. 2012).

In the first round, various versions of the dose response

are tested which are shown schematically in the flow chart

of Fig. 1. To retain clarity, not all tested models are shown.

A second round would involve improvements with dose

effect modifications by explanatory variables such as sex,

age at exposure or attained age, an example is given in

Eq. (4). After passing an LRT, a model is kept for further

rounds of testing. It may join Occam’s group, if improve-

ments are no longer possible. Defeated models are rejected

for risk assessment. In Fig. 1, a defeated model is identified

by at least one arrow pointing away from it. Models sur-

viving the last round of tests ‘see’ only arrowheads. More

details of the protocol are given in Kaiser et al. (2012).

In the present analysis, an additional selection criterion

prevents the overpopulation of Occam’s group with models

Radiat Environ Biophys (2013) 52:17–27 19

123

of negligible influence. Based on the Akaike Information

(Akaike 1973; Burnham and Anderson 2002)

AIC ¼ devþ 2 Npar; ð2Þ

where dev denotes the Poisson deviance and Npar denotes the

number of parameters, a weight 1=½1þ expð�DAICk=2Þ�can be constructed for the pairwise comparison of the pre-

ferred model with AIC0 and model k with AICk, where

DAICk ¼ AICk � AIC0: If this weight fell below 5 % (or

DAICk exceeds 5.99), the corresponding model k was not

used for risk assessment (Hoeting et al. 1999; Walsh 2007).

Note, that the second criterion does not constitute a statistical

test (Burnham and Anderson 2002, p 84). After its applica-

tion, the L-exp model with dose effect modifier for attained

age has been discarded (see Fig. 1).

At the end of the selection procedure, Occam’s group of

non-nested risk models with enough relevance for risk

assessment has been established for use in the MMISP

analysis.1

Candidate models for Occam’s group

From the outset, the dose response of candidate models is

constrained to yield a zero excess risk at zero dose and to

rise monotonously with an increasing dose. Models with

hormetic dose responses have not been tested but would

have been admitted into Occam’s group if they qualified.

Apart from these preconditions, admission to Occam’s

group is achieved solely by sufficient goodness-of-fit.

Improvements of the baseline model from Eq. (1) have

been attempted with three types of dose responses ‘LQ-

exp’, ‘sigmoid’ and ‘spline’ (see Fig. 1) for both EAR and

ERR models. The complete dose response of the LQ-exp

model took the form (a d ? b d2)exp(-c d). To account

for random errors, the dose-squared covariable has been

multiplied with a factor of 1.12 (Walsh and Kaiser 2011;

Pierce et al. 1990). Sub-models with all seven possible

combinations of the dose–response parameters a, b and chave been tested but only the two parameter combinations

a, b (sub-model LQ for linear-quadratic) and b, c (sub-

model Q-exp for quadratic-exponential) survived the series

of LRTs. Cubic-exponential or quadratic-exponential

models did not yield better fits than the Q-exp model. But a

model with a sigmoidal response (which progresses from

small beginnings and levels off at high doses) and a model

with two linear dose responses, connected by a break point

at dose dk (termed spline model), could also be added to

Occam’s group.

To perform valid LRTs, two continuous derivatives

(i.e. a C2 condition) of the Poisson deviance with respect

to the model parameters are required (Schervish 1997).

All but one model apply parametric functions which are

twice continuously differentiable. For the spline model, it

is not obvious that the C2 condition is fullfilled for

derivatives with respect to the break point dk. Therefore,

the region around the minimum of the Poisson likelihood

as a function of dk has been scanned numerically by

fixing dk at different values and re-fitting the remaining

parameters. The scan revealed a slightly tilted paraboloid

so that both derivatives are indeed continuous. The min-

imum is reached at dk = 0.36Sv (r CILP 0.28; 0.52). The

CILP are calculated from the likelihood profile with the

LQ

L Q

sigmoid splineae

a ae e

e a

ae

11

10

7

9

8

4.02 0.007.02

2.021.89

bsl

Npar

pxe−Qpxe−L

Fig. 1 Flow chart of model selection. Models are grouped in rows

pertaining to equal number of model parameters Npar. The protocol

starts with the baseline model bsl (top), arrows point to models which

survived a pairwise LRT on the 95 % level. Dose effect modifiers are

annotated as e for age at exposure and as a for attained age. AIC

differences to the preferred model Q-exp with dose effect modifier for

a are given for all models surviving the last round of tests. Model

L-exp with dose effect modifier for a is discarded because its DAICexceeded 5.99 (dashed arrow line)

1 The present study is named MMISP study, since models are chosen

by a Selection Protocol. For a better distinction, the study of Walsh

and Kaiser (2011) is named here MMIPM study, since it was based on

previously Published Models.

20 Radiat Environ Biophys (2013) 52:17–27

123

MINOS routine of MINUIT2. A graphical evaluation of

the numerical scan yielded the same values.

Dose effect modifiers of sex s, age at exposure e and

attained age a have been tested separately and in combi-

nation but only the modifier exp e ln a55

� �has been accepted

in all three types of dose responses shown in Fig. 1. The

difference between males and females was not significant

for all selected ERR models in contrast to the results of the

(discarded) EAR models.

Determination of model-specific risk estimates

and confidence intervals

A best risk estimate for a single model is calculated with

the set of parameter estimates which minimises the likeli-

hood. To determine the corresponding CI, a probability

density function (pdf) with 10,000 entries is generated by

Monte-Carlo simulation which accounts for uncertainty

ranges and pairwise correlations of all adjusted parameters.

Two percentiles, corresponding to the required level of

confidence (i.e. 95 %), are adopted as upper and lower CI.

To meet the requirement of a symmetric parameter

correlation matrix as the backbone of the Monte-Carlo

simulation, each parameter-specific pdf must ideally follow

a Gaussian distribution. As a necessary precondition, the rCILP, calculated from the likelihood profile, should lie

symmetrically around the best parameter estimate. The

precondition is fullfilled for the baseline model given in

Eq. (1) which is used by the models of Occam’s group. All

parameters of the ERR in the LQ and the Q-exp model

show symmetric CILP to a good approximation if models

are centred at e = 30 and a = 55 (see Table 1). However,

models centred at young ages at exposure and attained ages

exhibit markedly skewed CILP for the linear and the qua-

dratic term in the ERR(d). The ERR parameters a and b of

the sigmoid model possess asymmetric CILP with ratios of

0.7 and 2 between lower and upper bound but the asym-

metry did not disappear for centring at different values.

The spline model had symmetric CILP for the two linear

risk coefficients but the break point dk showed asymmetric

CILP for all tested combinations of centring. To calculate

CI with Monte-Carlo simulation for all five combinations

of e and a, the models have been centred at e = 30 and

a = 55. Although the precondition of symmetric parameter

CILP is not fully met for two ERR parameters of the sig-

moid model and one parameter of the spline model, one

expects that Monte–Carlo simulations of uncertainties for

these two models yield results with a moderate bias.

Centring does not change the quality of fit, i.e., the value

of the Poisson deviance and the best risk estimates. Walsh

and Kaiser (2011) exploited this fact and centred the risk

models at seven pairs of a and e for a more convenient

calculation of uncertainties. Especially at young ages, their

approach (implemented in their Method 1) yielded sym-

metric CI in the Monte-Carlo simulations even if the correct

CILP from the profile likelihood were highly asymmetric.

To partly make up for this bias, the simulation of CI in their

MMIPM analysis has been repeated with their models cen-

tred at e = 30 and a = 55 with approx. symmetric CILP.

Moreover, the complete parameter correlation matrix was

used now to simulate parameter uncertainties instead of the

fraction that pertained to the ERR part of the model. In the

repeated analysis, only the four models with the highest

weights (see Table 3 of Walsh and Kaiser (2011)) were

applied to the data set of LSS report 14 (Ozasa et al. 2012).

Now the 2.5 % percentiles of the ERR for the UNSCEAR

model do not drop below zero in contrast to the results

reported in Table 4 of Walsh and Kaiser (2011).

Multi-model inference

The surviving models are ranked according to their AIC,

defined in Eq. (2), and to each model k an AIC-related

weight

pk ¼expð� 1

2DAICkÞ

PM�1j¼0 expð� 1

2DAICjÞ

ð3Þ

has been assigned.

The central risk estimate from MMI is given by the AIC-

weighted mean of best estimates from the models in

Occam’s group. The CI of the MMI mean are derived from

a joint pdf with 10,000 entries which is obtained by merging

the model-specific pdf with sizes corresponding to the AIC-

weight (i.e. 5,301, 2,062, 1,927 and 710 realisations from

models Q-exp, sigmoid, spline and LQ, see Table 2). From

the joint pdf, an approximation of the unconditional sam-

pling variance [see Burnham and Anderson 2002, Eq. (4.3)]

Table 1 Best parameter estimates, symmetric Wald-type DCI from a

parabolic approximation around the minimum of the likelihood

function, and DCILP from the actual likelihood profile for the pre-

ferred Q-exp model; to facilitate the assessment of symmetry, DCIare

given as distances from the best parameter estimate in the standard rrange

Name Unit Best estim. Wald-type DCI DCILP

b0 – -9.50 0.10 -0.11; 0.10

bs – -0.322 0.057 -0.057; 0.057

bc – -0.140 0.065 -0.066; 0.064

ba1– 2.11 0.27 -0.27; 0.27

ba2– 1.08 0.21 -0.21; 0.20

be110-3 year-1 6.4 -0.46 -0.46; 0.46

be210-4 year-2 -7.2 2.3 -2.3; 2.2

b Sv-2 4.3 1.2 -1.1; 1.4

c Sv-1 -0.38 0.13 -0.13; 0.13

� – -1.62 0.35 -0.36; 0.34

Radiat Environ Biophys (2013) 52:17–27 21

123

can be obtained. Implicitly, this pdf also accounts for model

correlations.

Results

For a total of 26 ERR and 16 EAR candidate models, lists

of Poisson deviances, number of parameters and AIC val-

ues are given in the online resource as a PDF excerpt of an

EXCEL workbook (ESM1). The AIC of the preferred EAR

model was still about 11 points away from the AIC of the

preferred ERR model. Thus, no EAR model fell within

Occam’s group.

For the four selected models, files with model-specific

data on the quality of fit, parameter estimates and CI (from

both the parabolic approximation of the likelihood mini-

mum and from the likelihood profile), the parameter cor-

relation matrix and tables to compare observed and

expected cases are added to the online resource in PDF

format. The data provided allow a repetition of the MMISP

analysis without re-fitting the corresponding models.

Table 2 presents the four ERR models in Occam’s

group. Only the dose dependence ERR(d) is shown there,

the final form

ERRðd; aÞ ¼ ERRðdÞ exp e lna

55

� �ð4Þ

additionally applies a power function for attained age

a centred at 55 year.

Compared to the previous analysis, the baseline function

of both the LQ model and LQ-exp model from Schneider and

Walsh (2009) was replaced by Eq. (1) with one parameter

less which increased the deviance by only about one point.

Accounting for the explanatory variables of sex and age at

exposure yielded no significant improvements of their

models so they were discarded. With these modifications, the

LQ model of Schneider and Walsh (2009) morphed into the

LQ model of the present analysis, which is equivalent to

the UNSCEAR model considered in Walsh and Kaiser

(2011). With the same modifications, and after elimination

of the linear term, the LQ-exp model of Schneider and Walsh

(2009) became the preferred Q-exp model of the present

analysis with parameter estimates given in Table 1. The

model of Little et al. (2008) was excluded from Occam’s

group because the dependence on age at exposure did not

survive the LRT with the UNSCEAR model.

The UNSCEAR model (termed LQ model in the present

analysis) dominated the MMIPM risk estimate in Walsh and

Kaiser (2011) with a weight of 51 % (see their Table 5),

but here its contribution is reduced to only 7 %. Now the

Q-exp model is preferred with a weight of 53 % with a four

points lower deviance than the LQ model. Inspection of

tables, which compare observed and expected cases in

model-specific result files (here ESM3 and ESM2) of the

online resource, suggests that the Q-exp model produced

slightly better fits to the data at young ages of exposure and

attained ages. For example, for Poisson strata (numbered 0,

10, 20 in the result files) with person-year weighted means

of e ^ 5 year, a ^ 15 y the contribution to the Poisson

deviance of the Q-exp model is about 2.5 points lower

compared to the LQ model. Such exposure scenarios are

of enhanced interest for radiation protection and here

the Q-exp model yields lower (and better supported)

risk estimates than models with a linear-quadratic dose

response.

The quadratic term of the Q-exp model determines the

response at doses \0.5 Sv, damping by the exponential

term becomes important above [2.5 Sv. In the intermedi-

ate range between 0.5 and 2.5 Sv, the response is well

approximated by a linear relationship (see Fig. 2). Between

2.5 and 3 Sv, nine cases have been recorded and there are

only two cases above 3 Sv. The markedly different risk

estimates for high doses are caused by the low statistical

power in the corresponding cohort strata.

Table 2 Parametric dose dependence for the ERR models of

Occam’s group used in MMI, the AIC-weight is calculated from

Eq. (3)

Model

name

Form of ERR(d) Npar Deviance AIC-

Weight

Q-exp bd2 expðcdÞ 10 2,670.890 0.5301

Sigmoid ABþdC 11 2,670.778 0.2062

Spline a1d for d\dk

a2ðd � dkÞ for d� dk

�11 2,670.914 0.1927

LQ a d ? b d2 10 2,674.910 0.0710

0 1 2 3 4

Bone Marrow Dose (Sv)

0

10

20

30

Exc

ess

Rel

ativ

e R

isk

MMISP (present analysis)

Q-expsigmoidsplineLQMMIPM (Kaiser & Walsh 2011)

Fig. 2 Excess relative risk (AIC-weighted mean or best estimate for

separate models, with 95 % CI) for a 55-year old adult exposed at age

30 from MMISP (present analysis), the preferred Q-exp model, the

sigmoid model, the spline model, the LQ model and from the repeated

MMIPM analysis with the four top-ranking models of Walsh and

Kaiser (2011)

22 Radiat Environ Biophys (2013) 52:17–27

123

The ERR at low doses for a 7-year-old child exposed at

age 2 is shown in Fig. 3. Compared to the previous anal-

ysis, the AIC-weighted mean of the ERR from MMISP is

reduced, i.e.,. by a factor of two at 100 mSv, although the

reduction is not statistically significant. The effect of any

one model is directly visible in the MMI dose response if it

has a weight of more than fifty per cent. The AIC-weighted

mean from MMISP closely follows the best estimate of the

preferred Q-exp model. The additional three models cause

a sizeable increase of the CI especially at low doses where

a determination of the ERR implies an extrapolation to

cohort strata with almost no cases (see Table 2 of Walsh

and Kaiser 2011). In these regions, CI from a single model

of choice underestimate the risk uncertainty by wide mar-

gins (see also Tables 3, 4).

Tables 3, 4 and 5 present the ERR from the four models of

Occam’s group separately and from MMISP of the present

analysis and of the MMIPM analysis by Walsh and Kaiser

(2011) at 10 mSv, 100 mSv and 1 Sv. At exposure of 1 Sv,

both MMI analyses and all separate models yield similar

estimates and CI for children, adolescents and adults.

The situation changes at 100 mSv. Now the new pre-

ferred Q-exp model predicts a four times lower risk than

the previously chosen UNSCEAR (here LQ) model.

Compared to the repeated MMIPM analysis with the four

top-ranking models of Walsh and Kaiser (2011), estimates

from the present MMISP analysis differ by a factor of 2.5

and the CI are markedly reduced.

At 10 mSv, the AIC-weighted mean of the present study

no longer approximates the best estimate of the preferred

Q-exp model. The mean is strongly influenced by a 30

times higher estimate of the LQ model which on the other

hand acquires the lowest weight in MMISP. To avoid this

effect and to preserve the similarity between the point

estimates from the preferred model and from MMI, Kaiser

et al. (2012) recommend to replace the AIC-weighted

mean by the median of the joint pdf, which is given in

brackets in Table 3.

At doses of 10 mSv and 100 mSv, the 2.5 % percentiles

from the present MMISP analysis include a zero risk due to

the uncertainty of the spline model.

0 0.1 0.2 0.3

Bone Marrow Dose (Sv)

-5

0

5

10

15

20E

xces

s R

elat

ive

Ris

k

MMISP (present analysis)

Q-expMMIPM (Walsh & Kaiser 2011)

Fig. 3 Excess relative risk (AIC-weighted mean or best estimate for

model Q-exp, with 95 % CI) for a 7-year old child exposed at age 2

from MMISP (present analysis), the preferred Q-exp model and from

the repeated MMIPM analysis with the four top-ranking models of

Walsh and Kaiser (2011)

Table 3 ERR (10-2) at 10 mSv

for five combinations of age at

exposure e and attained age a

AIC-weighted mean for MMI or

model-specific best estimate in

first row, 95 % CI from Monte-

Carlo simulation in second row,

for MMISP the mean is

calculated with the model-

specific weights of Table 2

y median of joint pdf from

MMISP in brackets� Point estimates and CI from

repeated analysis (see text)

Model name or

MMI result

e = 2

a = 7

e = 2

a = 12

e = 7

a = 17

e = 12

a = 22

e = 30

a = 55

MMIPM 33.5 13.7 7.80 5.12 1.14

Walsh and

Kaiser (2011)*

-33.5; 208 -12.4; 61.7 -7.24; 28.5 -4.78; 16.4 -1.22; 2.77

LQ 40.4 16.8 9.48 6.22 1.39

UNSCEAR

(2008)

-0.753; 215 -0.370; 64.0 -0.237; 29.2 -0.167; 16.5 -0.0402; 2.80

Spline 17.7 7.34 4.15 2.72 0.609

-36.3; 153 -12.9; 46.1 -6.98; 22.0 -4.36; 12.8 -1.00; 2.19

Sigmoid 0.851 0.358 0.204 0.135 0.0310

1.2 9 10-3;

41.2

5.5 9 10-4;

14.9

3.2 9 10-4;

7.92

2.2 9 10-4;

5.06

5.8 9 10-5;

1.04

Q-exp 1.20 0.501 0.285 0.188 0.0427

(preferred

model)

0.246; 5.02 0.140; 1.50 0.0953; 0.699 0.0696; 0.398 0.0184; 0.0667

MMIySP

7.09 (1.41) 2.94 (0.568) 1.67 (0.317) 1.09 (0.206) 0.245 (0.0463)

(present study) -9.02; 92.7 -3.67; 31.8 -2.01; 16.3 -1.34; 9.89 -0.323; 1.99

Radiat Environ Biophys (2013) 52:17–27 23

123

Discussion

Little et al. (1999) analysed the dose response for three

subtypes of acute myeloid leukaemia (AML), chronic

myeloid leukaemia (CML) and acute lymphocytic leukae-

mia (ALL) separately and for all subtypes combined. Their

analysis was carried out with LSS incidence data, and with

two other data sets of women treated for cervical cancer

(incidence) and UK patients treated for ankylosing

spondilitis (mortality). From a list of 13 ERR models, using

similar versions of the general LQ-exp response with dose

effect modifiers, the Q-exp response has been preferred for

yielding the optimal fit. They used already LRTs to discard

models with statistically insignificant features. Their esti-

mates of the coefficients b for the dose squared and c for the

exponential damping were 5.8 (95 % CI 2.7; 11) Sv-2 and

-0.49 (95 % CI -0.76; -0.22) Sv-1, respectively (see

their Table 5). Risk estimates for leukaemia incidence are

expected to exceed those for mortality. Comparison with

estimates in Table 1 shows that this relation is realised for

dose . 3 Sv, albeit without statistical significance.

Separate estimates for the other two data sets produced

no significant risk (women with cervical cancer) or a ten

times larger coefficient b (patients with ankylosing

spondilitis). Comparison of risks in these different popula-

tions is complicated by the consideration that the LSS

subjects were not under observation because of known

diseases whereas members of the two other data sets were.

Basic tenets of MMI might be extended to address

questions of risk transfer between populations which are

discussed in reports of committees BEIR VII (NRC 2006)

and ICRP (Valentin 2007). BEIR VII propose to transfer

risks for solid cancer sites (except breast and thyroid) and

for leukaemia from the LSS cohort to the US population

with a linear combination of an ERR model and an EAR

model. They recommend point estimates as weighted means

obtained under the two models. For leukaemia and solid

cancer sites (except breast, thyroid and lung), the weights of

Table 5 ERR at 1 Sv for five

combinations of age at exposure

e and attained age a

AIC-weighted mean for MMI or

model-specific best estimate in

first row, 95 % CI from Monte-

Carlo simulation in second row,

for MMISP the mean is

calculated with the model-

specific weights of Table 2

* Point estimates and CI from

repeated analysis (see text)

Model name or

MMI result

e = 2

a = 7

e = 2

a = 12

e = 7

a = 17

e = 12

a = 22

e = 30

a = 55

MMIPM 81.8 32.9 18.7 12.3 2.76

Walsh and Kaiser (2011)* 17.0; 356 10.2; 97.7 6.24; 44.2 5.28; 24.9 1.60; 3.90

LQ 78.6 32.6 18.4 12.1 2.71

UNSCEAR (2008) 17.2; 348 10.2; 100 7.16; 44.9 5.48; 24.8 1.59; 3.81

Spline 101 42.0 23.8 15.6 3.49

22.8; 423 13.8; 121 9.83; 54.2 7.60; 30.1 2.27; 4.69

Sigmoid 91.1 38.3 21.9 14.5 3.31

18.7; 345 10.9; 102 7.61; 46.5 5.77; 26.3 1.56; 4.37

Q-exp (preferred model) 82.4 34.4 19.6 12.9 2.94

(preferred model) 18.2; 322 10.8; 94.4 7.45; 43.1 5.55; 24.3 1.54; 3.91

MMISP (present study) 87.6 36.6 20.8 13.7 3.10

(present study) 19.7; 343 11.4; 101 7.95; 46.2 5.94; 25.9 1.60; 4.31

Table 4 ERR (10-1) at

100 mSv for five combinations

of age at exposure e and

attained age a

AIC-weighted mean for MMI or

model-specific best estimate in

first row, 95 % CI from Monte-

Carlo simulation in second row,

for MMISP the mean is

calculated with the model-

specific weights of Table 2

* Point estimates and CI from

repeated analysis (see text)

Model name or

MMI result

e = 2

a = 7

e = 2

a = 12

e = 7

a = 17

e = 12

a = 22

e = 30

a = 55

MMIPM 38.5 15.7 8.92 5.86 1.31

Walsh and Kaiser (2011)* -15.5; 219 -6.19; 64.7 -3.61; 29.8 -2.49; 17.0 -0.642; 2.84

LQ 43.9 18.2 10.3 6.76 1.51

UNSCEAR (2008) 2.37; 227 1.18; 66.5 0.745; 30.3 0.513; 17.2 0.125; 2.88

Spline 17.7 7.34 4.15 2.72 0.609

-36.3; 153 -12.9; 46.1 -6.98; 22.0 -4.36; 12.8 -1.00; 2.19

Sigmoid 9.58 4.03 2.30 1.52 0.348

0.298; 91.9 0.145; 31.4 0.0889; 15.7 0.0620; 9.56 0.0164; 1.89

Q-exp (preferred model) 11.6 4.84 2.76 1.82 0.413

(preferred model) 2.39; 48.2 1.37; 14.4 0.932; 6.67 0.683; 3.79 0.182; 0.633

MMISP 14.6 6.10 3.47 2.28 0.515

(present study) -8.60; 103 -3.58; 34.0 -1.97; 17.6 -1.31; 10.5 -0.312; 2.07

24 Radiat Environ Biophys (2013) 52:17–27

123

0.7 (ERR) and 0.3 (EAR) are chosen by expert judgement

based on the observation ‘that there is a somewhat greater

support for relative risk than for absolute risk transport’ (see

p. 276). Inconsistent with BEIR VII, ICRP recommend to

apply only the EAR model of Preston et al. (1994) for

leukaemia incidence.

In general, the consensus on a risk transfer model is based

on a complex mix of factors, but a comprehensive consid-

eration is beyond the scope of the present study. However,

any adequate transfer model should provide a good

description of the risk in the population of origin. Would

goodness-of-fit criteria be allowed to assess the adequacy of

a model, EAR models of leukaemia mortality would not

contribute to the transfer. The best EAR model exceeds the

AIC of the preferred Q-exp model by about 11 points which

leads to a negligible AIC-weight. A second criterion of

Bayesian information ðBIC ¼ devþ Npar lnðncasesÞÞ is often

used as an alternative to the AIC because it favours more

parsimonious models (Claeskens and Hjort 2008). It is 18

points higher which constitutes strong evidence (Kass and

Raftery 1995) for the rejection of the EAR model. Likewise,

Little (2008) recommends to drop EAR models, but with a

different rationale. Based on a comparison of risks for

childhood exposure between the LSS cohort and three

medically exposed groups in Europe, he observed that het-

erogeneity in cohort-specific EAR estimates is much higher

than in ERR estimates.

A recent risk study of leukaemia (and brain tumours)

after childhood exposure by CT scans reports an ERR of 36

(95 % CI 5; 120) Sv-1 from a purely linear model for age at

exposure\22 year, dose range between 0 and 100 mSv and

follow-up of 23 year (Pearce et al. 2012). The same linear

model applied to the LSS incidence data (Preston et al.

1994) produced an ERR of 37 (95 % CI 14; 127) Sv-1 for

age at exposure \20 year, dose range between 0 and 4 Sv

and follow-up of 20 year (see Table 8 of the supplement to

Pearce et al. (2012)).

The authors of the present study fitted a purely qua-

dratic model to the LSS incidence data for all dose ranges

which increased the deviance by 4.8 points compared to

the linear model. If the overlap of dose ranges was

improved by a reduction to 0–500 mSv for the LSS data,

the quadratic model yielded a slightly better fit. The

deviance decreased by 2.4 points compared to the linear

model. Improved fits of a quadratic model at lower doses

are in line with findings of the present study (mortality)

and the study of Little et al. (1999) (incidence). With a

coefficient of 61 (95 % CI 22; 185) Sv-2 for the quadratic

model, the ERR at 100 mSv is six times lower than for the

linear model. Using both models in MMI would still yield

a reduction of the ERR by a factor of three compared to

the linear model.

Nevertheless, Pearce et al. (2012) report ‘little evidence

of nonlinearity of the dose response, using either linear-

quadratic or linear-exponential forms of departure from

linearity’, but purely quadratic responses appear not to

have been tested. At this point, the present authors suggest

a comparison of the CT risk with the LSS quadratic

response. Should alternative dose responses, such as purely

quadratic, fit the data comparably well, an even fairer

comparison might account for model uncertainty in the CT

cohort. In this case, reverting to MMI can relieve the

dilemma of needing to choose between models with largely

different consequences for issues of public health, e.g., for

assessing the risk-to-benefit ratio related to a CT scan. In a

wider context, MMI might be of use for statistical analysis

in a number of cohort studies of CT exposure and cancer

incidence which will be completed in the near future

(Einstein 2012).

Conclusions

Only models with a linear-quadratic dose response were

included in the MMI analysis of Walsh and Kaiser (2011).

The present analysis introduced three models with non-

standard dose responses which produced significantly bet-

ter fits to the data. All considered models yield very similar

point estimates and uncertainties in the dose range

0.5–2.5 Sv, i.e., in cohort strata with a sufficient number of

cases. Divergent predictions appear in strata with almost no

cases for children and adolescents exposed to very low

doses of 100 mSv and below (see Table 2 of Walsh and

Kaiser 2011). Yet for purposes of radiation protection,

these exposure scenarios are of increased interest. Com-

pared to the study of Walsh and Kaiser (2011), the present

MMI analysis predicts markedly lower risks with factors of

two around 100 mSv and up to five for lower doses. These

point estimates are considered as more reliable since they

were produced with models, which describe the data

slightly better notably for children and adolescents.

Besides the improvement of point estimates, a second

benefit of MMI has been demonstrated. Several plausible

models can be included in a more comprehensive (though

not exhaustive) determination of uncertainties. Again, the

benefit becomes noticeable in the above-mentioned cohort

strata with low statistical power, where the risk is deter-

mined by extrapolation. Now uncertainty ranges are mainly

determined by the spread of model-specific point esti-

mates, whereas the model-specific uncertainty ranges are

rather small. Hence, inferring uncertainties from a single

model of choice may lead to a substantial underestimation.

In this context, the present MMI study provides significant

risks only above some three hundred mSv, whereas the

Radiat Environ Biophys (2013) 52:17–27 25

123

95 % CI of the preferred Q-exp model do not include a

zero risk for all considered exposure scenarios.

The impact of pertinent sources of uncertainty, such

as the ‘healthy survivor effect’, errors in dosimetry or

misdiagnosis of cases on risk estimates has been discussed

extensively in the literature, for the LSS cohort see, e.g.,

Little et al. (1999), Preston et al. (2003), Preston et al.

(2004). Already Little et al. (1999) preferred ERR models

with a Q-exp response. They did, however, not consider the

additional contribution to the uncertainty which is induced

by including models with other plausible dose responses

into the risk analysis. In this developing field of research in

radiation epidemiology, the present MMI study aims to be

of help.

The model selection bias cannot be eliminated by MMI

but can be markedly reduced. The bias is transferred from

the level of picking a single model of choice to picking a set

of candidate models. In the present analysis, this set inclu-

ded more than 40 models with different forms of dose

responses, of which four models have been admitted to

Occam’s group. Under the given rules for model selection, it

appears unlikely that by broadening the basis of candidate

models a considerable number of new models would enter

Occam’s group. Even if new models appeared, their impact

on risk estimates would be contained by the original models.

Acknowledgments This study makes use of data obtained from the

Radiation Effects Research Foundation (RERF) in Hiroshima and

Nagasaki, Japan. RERF is a private, non-profit foundation funded by

the Japanese Ministry of Health, Labour and Welfare (MHLW) and

the U.S. Department of Energy (DOE), the latter in part through the

National Academy of Sciences. The data include information

obtained from the Hiroshima City, Hiroshima Prefecture, Nagasaki

City, and Nagasaki Prefecture Tumor Registries and the Hiroshima

and Nagasaki Tissue Registries. The conclusions in this study are

those of the authors and do not necessarily reflect the scientific

judgement of RERF or its funding agencies.

Open Access This article is distributed under the terms of the

Creative Commons Attribution License which permits any use, dis-

tribution, and reproduction in any medium, provided the original

author(s) and the source are credited.

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